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Improving Quantum Machine Learning via Heat-Bath Algorithmic Cooling
PRX Quantum 7, 010350 – Published 12 March, 2026
DOI: https://doi.org/10.1103/24n3-tskj
Abstract
This work introduces an approach rooted in quantum thermodynamics to enhance sampling efficiency in quantum machine learning (QML). We propose conceptualizing quantum supervised learning as a thermodynamic cooling process. Building on this concept, we develop a quantum refrigerator protocol that enhances sample efficiency during training and prediction without the need for Grover iterations or quantum phase estimation. Inspired by heat-bath algorithmic cooling protocols, our method alternates entropy compression and thermalization steps to decrease the entropy of qubits, increasing polarization toward the dominant bias. This technique minimizes the computational overhead associated with estimating classification scores and gradients, presenting a practical and efficient solution for QML algorithms compatible with noisy intermediate-scale quantum devices.
Physics Subject Headings (PhySH)
synopsis
Cool Qubits Make Faster Decisions
Applying thermodynamic principles makes quantum machine-learning protocols more efficient.
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Popular Summary
Quantum machine learning (QML) algorithms rely on repeated measurements, which can make them slow and resource-intensive on quantum hardware. This sampling overhead is a major bottleneck for using quantum classifiers in practice. To address this, we draw inspiration from quantum thermodynamics, treating learning as a cooling process. Based on this idea, we develop bidirectional quantum refrigerators that amplify the signal used by a quantum classifier to distinguish different classes, without knowing the correct answer in advance. This reduces the number of measurements needed for both training and prediction, while remaining compatible with near-term quantum devices. Our method works directly on quantum states and applies broadly to variational quantum classifiers and related models. More broadly, this work shows how thermodynamic ideas can offer tools for data processing in QML, opening fresh directions for making quantum algorithms more practical and scalable.
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References (82)
- Jacob Biamonte, Peter Wittek, Nicola Pancotti, Patrick Rebentrost, Nathan Wiebe, and Seth Lloyd, Quantum machine learning, Nature 549, 195 (2017).
- Vojtech Havlícek, Antonio D. Córcoles, Kristan Temme, Aram W. Harrow, Abhinav Kandala, Jerry M. Chow, and Jay M. Gambetta, Supervised learning with quantum-enhanced feature spaces, Nature 567, 209 (2019).
- Yunchao Liu, Srinivasan Arunachalam, and Kristan Temme, A rigorous and robust quantum speed-up in supervised machine learning, Nat. Phys. 17, 1013 (2021).
- Jennifer R. Glick, Tanvi P. Gujarati, Antonio D. Corcoles, Youngseok Kim, Abhinav Kandala, Jay M. Gambetta, and Kristan Temme, Covariant quantum kernels for data with group structure, Nat. Phys. 20, 479 (2024).
- Hsin-Yuan Huang, Michael Broughton, Jordan Cotler, Sitan Chen, Jerry Li, Masoud Mohseni, Hartmut Neven, Ryan Babbush, Richard Kueng, John Preskill, and Jarrod R. McClean, Quantum advantage in learning from experiments, Science 376, 1182 (2022).
- M. Cerezo, Guillaume Verdon, Hsin-Yuan Huang, Lukasz Cincio, and Patrick J. Coles, Challenges and opportunities in quantum machine learning, Nat. Comput. Sci. 2, 567 (2022).
- Edward Farhi and Hartmut Neven, Classification with quantum neural networks on near term processors, arXiv:1802.06002.
- Jarrod R. McClean, Sergio Boixo, Vadim N. Smelyanskiy, Ryan Babbush, and Hartmut Neven, Barren plateaus in quantum neural network training landscapes, Nat. Commun. 9, 4812 (2018).
- Iris Cong, Soonwon Choi, and Mikhail D. Lukin, Quantum convolutional neural networks, Nat. Phys. 15, 1273 (2019).
- M. Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C. Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R. McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, and Patrick J. Coles, Variational quantum algorithms, Nat. Rev. Phys. 3, 625 (2021).
- Gilles Brassard, Peter Hoyer, Michele Mosca, and Alain Tapp, Quantum amplitude amplification and estimation, Contemp. Math. 305, 53 (2002).
- Ashley Montanaro, Quantum speedup of Monte Carlo methods, Proc. R. Soc. A: Math. Phys. Eng. Sci. 471, 20150301 (2015).
- Yohichi Suzuki, Shumpei Uno, Rudy Raymond, Tomoki Tanaka, Tamiya Onodera, and Naoki Yamamoto, Amplitude estimation without phase estimation, Quantum Inf. Process. 19, 1 (2020).
- Dmitry Grinko, Julien Gacon, Christa Zoufal, and Stefan Woerner, Iterative quantum amplitude estimation, npj Quantum Inf. 7, 52 (2021).
- John Preskill, Quantum computing in the NISQ era and beyond, Quantum 2, 79 (2018).
- Kishor Bharti, Alba Cervera-Lierta, Thi Ha Kyaw, Tobias Haug, Sumner Alperin-Lea, Abhinav Anand, Matthias Degroote, Hermanni Heimonen, Jakob S. Kottmann, Tim Menke, Wai-Keong Mok, Sukin Sim, Leong-Chuan Kwek, and Alán Aspuru-Guzik, Noisy intermediate-scale quantum algorithms, Rev. Mod. Phys. 94, 015004 (2022).
- P. Oscar Boykin, Tal Mor, Vwani Roychowdhury, Farrokh Vatan, and Rutger Vrijen, Algorithmic cooling and scalable NMR quantum computers, Proc. Natl. Acad. Sci. U.S.A. 99, 3388 (2002).
- Leonard J. Schulman and Umesh V. Vazirani, Molecular scale heat engines and scalable quantum computation, in Proceedings of the Thirty-First Annual ACM Symposium on Theory of Computing, STOC ’99 (Association for Computing Machinery, New York, NY, USA, 1999), pp. 322–329.
- Daniel K. Park, Nayeli A. Rodriguez-Briones, Guanru Feng, Robabeh Rahimi, Jonathan Baugh, and Raymond Laflamme, Heat Bath Algorithmic Cooling with Spins: Review and Prospects (Springer, New York, 2016), p. 227.
- Yuval Elias, Tal Mor, and Yossi Weinstein, Semioptimal practicable algorithmic cooling, Phys. Rev. A 83, 042340 (2011).
- Daniel K. Park, Guanru Feng, Robabeh Rahimi, StÉphane Labruy‘ere, Taiki Shibata, Shigeaki Nakazawa, Kazunobu Sato, Takeji Takui, Raymond Laflamme, and Jonathan Baugh, Hyperfine spin qubits in irradiated malonic acid: Heat-bath algorithmic cooling, Quantum Inf. Process. 14, 2435 (2015).
- Nayeli Azucena Rodríguez-Briones, Novel heat-bath algorithmic cooling methods, Ph.D. thesis, University of Waterloo, 2020.
- Junan Lin, Nayeli A. Rodríguez-Briones, Eduardo Martín-Martínez, and Raymond Laflamme, Thermodynamic analysis of algorithmic cooling protocols: Efficiency metrics and improved designs, Phys. Rev. A 110, 022215 (2024).
- Ralph Silva, Pharnam Bakhshinezhad, and Fabien Clivaz, Optimal unitary trajectories under commuting target and cost observables; applications to cooling, arXiv:2412.07291.
- Nayeli Azucena Rodríguez-Briones and Raymond Laflamme, Achievable polarization for heat-bath algorithmic cooling, Phys. Rev. Lett. 116, 170501 (2016).
- Lindsay Bassman Oftelie, Antonella De Pasquale, and Michele Campisi, Dynamic cooling on contemporary quantum computers, PRX Quantum 5, 030309 (2024).
- Christopher M. Bishop, Pattern Recognition and Machine Learning, Information Science and Statistics (Springer, New York, 2006).
- Shai Shalev-Shwartz and Shai Ben-David, Understanding Machine Learning: From Theory to Algorithms (Cambridge University Press, Cambridge, 2014).
- Roberto Giuntini, Federico Holik, Daniel K. Park, Hector Freytes, Carsten Blank, and Giuseppe Sergioli, Quantum-inspired algorithm for direct multi-class classification, Appl. Soft Comput. 134, 109956 (2023).
- Zoë Holmes, Kunal Sharma, M. Cerezo, and Patrick J. Coles, Connecting ansatz expressibility to gradient magnitudes and barren plateaus, PRX Quantum 3, 010313 (2022).
- Edward Grant, Marcello Benedetti, Shuxiang Cao, Andrew Hallam, Joshua Lockhart, Vid Stojevic, Andrew G. Green, and Simone Severini, Hierarchical quantum classifiers, npj Quantum Inf. 4, 65 (2018).
- Arthur Pesah, M. Cerezo, Samson Wang, Tyler Volkoff, Andrew T. Sornborger, and Patrick J. Coles, Absence of barren plateaus in quantum convolutional neural networks, Phys. Rev. X 11, 041011 (2021).
- Tak Hur, Leeseok Kim, and Daniel K. Park, Quantum convolutional neural network for classical data classification, Quantum Mach. Intell. 4, 3 (2022).
- Matt Lourens, Ilya Sinayskiy, Daniel K. Park, Carsten Blank, and Francesco Petruccione, Hierarchical quantum circuit representations for neural architecture search, npj Quantum Inf. 9, 79 (2023).
- Juhyeon Kim, Joonsuk Huh, and Daniel K. Park, Classical-to-quantum convolutional neural network transfer learning, Neurocomputing 555, 126643 (2023).
- M. Cerezo, Akira Sone, Tyler Volkoff, Lukasz Cincio, and Patrick J. Coles, Cost function dependent barren plateaus in shallow parametrized quantum circuits, Nat. Commun. 12, 1791 (2021).
- Nayeli A. Rodríguez-Briones, Jun Li, Xinhua Peng, Tal Mor, Yossi Weinstein, and Raymond Laflamme, Heat-bath algorithmic cooling with correlated qubit-environment interactions, New J. Phys. 19, 113047 (2017).
- Nayeli A. Rodríguez-Briones, Eduardo Martín-Martínez, Achim Kempf, and Raymond Laflamme, Correlation-enhanced algorithmic cooling, Phys. Rev. Lett. 119, 050502 (2017).
- Álvaro M. Alhambra, Matteo Lostaglio, and Christopher Perry, Heat-bath algorithmic cooling with optimal thermalization strategies, Quantum 3, 188 (2019).
- Philip Taranto, Patryk Lipka-Bartosik, Nayeli A. Rodríguez-Briones, Martí Perarnau-Llobet, Nicolai Friis, Marcus Huber, and Pharnam Bakhshinezhad, Efficiently cooling quantum systems with finite resources: Insights from thermodynamic geometry, Phys. Rev. Lett. 134, 070401 (2025).
- Tak Hur, Israel F. Araujo, and Daniel K. Park, Neural quantum embedding: Pushing the limits of quantum supervised learning, Phys. Rev. A 110, 022411 (2024).
- Carlos Ortiz Marrero, Mária Kieferová, and Nathan Wiebe, Entanglement-induced barren plateaus, PRX Quantum 2, 040316 (2021).
- Taylor L. Patti, Khadijeh Najafi, Xun Gao, and Susanne F. Yelin, Entanglement devised barren plateau mitigation, Phys. Rev. Res. 3, 033090 (2021).
- Jose M. Fernandez, Seth Lloyd, Tal Mor, and Vwani Roychowdhury, Algorithmic cooling of spins: A practicable method for increasing polarization, Int. J. Quantum Inf. 2, 461 (2004).
- Jonathan Baugh, Osama Moussa, Colm A. Ryan, Ashwin Nayak, and Raymond Laflamme, Experimental implementation of heat-bath algorithmic cooling using solid-state nuclear magnetic resonance, Nature 438, 470 (2005).
- C. A. Ryan, O. Moussa, J. Baugh, and R. Laflamme, Spin based heat engine: Demonstration of multiple rounds of algorithmic cooling, Phys. Rev. Lett. 100, 140501 (2008).
- Rodolfo R. Soldati, Durga B. R. Dasari, Jörg Wrachtrup, and Eric Lutz, Thermodynamics of a minimal algorithmic cooling refrigerator, Phys. Rev. Lett. 129, 030601 (2022).
- Yosi Atia, Yuval Elias, Tal Mor, and Yossi Weinstein, Algorithmic cooling in liquid-state nuclear magnetic resonance, Phys. Rev. A 93, 012325 (2016).
- Krishna Shende, Arvind, and Kavita Dorai, State-independent robust heat-bath algorithmic cooling of nuclear spins, Phys. Rev. Appl. 21, 024017 (2024).
- Xueyuan Hu and Valerio Scarani, Thermalization of finite complexity and its application to heat-bath algorithmic cooling, Phys. Rev. A 112, 042209 (2025).
- Jake Xuereb, Benjamin Stratton, Alberto Rolandi, Jinming He, Marcus Huber, and Pharnam Bakhshinezhad, Cooling a qubit using others, PRX Quantum 6, 040368 (2025).
- B. L. WELCH, The generalization of “STUDENT’S” problem when several different population varlances are involved, Biometrika 34, 28 (1947).
- Tak Hur and Daniel K. Park, Understanding Generalization in Quantum Machine Learning with Margins, in Proceedings of the 42nd International Conference on Machine Learning, Proceedings of Machine Learning Research, edited by Aarti Singh, Maryam Fazel, Daniel Hsu, Simon Lacoste-Julien, Felix Berkenkamp, Tegan Maharaj, Kiri Wagstaff, and Jerry Zhu (PMLR, Vancouver, Canada, 2025), Vol. 267, pp. 26338–26360, https://proceedings.mlr.press/v267/hur25a.html.
- Sofiene Jerbi, Lukas J. Fiderer, Hendrik Poulsen Nautrup, Jonas M. Kübler, Hans J. Briegel, and Vedran Dunjko, Quantum machine learning beyond kernel methods, Nat. Commun. 14, 517 (2023).
- Martin Larocca, Supanut Thanasilp, Samson Wang, Kunal Sharma, Jacob Biamonte, Patrick J. Coles, Lukasz Cincio, Jarrod R. McClean, Zoë Holmes, and M. Cerezo, A review of barren plateaus in variational quantum computing, Nat. Rev. Phys. 7, 174 (2025).
- Carsten Blank, Daniel K. Park, June-Koo Kevin Rhee, and Francesco Petruccione, Quantum classifier with tailored quantum kernel, npj Quantum Inf. 6, 1 (2020).
- Daniel K. Park, Carsten Blank, and Francesco Petruccione, Robust quantum classifier with minimal overhead, in 2021 International Joint Conference on Neural Networks (IJCNN) (2021), pp. 1–7.
- M. Schuld, M. Fingerhuth, and F. Petruccione, Implementing a distance-based classifier with a quantum interference circuit, Europhys. Lett. 119, 60002 (2017).
- Daniel K. Park, Carsten Blank, and Francesco Petruccione, The theory of the quantum kernel-based binary classifier, Phys. Lett. A 384, 126422 (2020).
- Carsten Blank, Adenilton J. da Silva, Lucas P. de Albuquerque, Francesco Petruccione, and Daniel K. Park, Compact quantum kernel-based binary classifier, Quantum Sci. Technol. 7, 045007 (2022).
- Nicolas M. de Oliveira, Daniel K. Park, Israel F. Araujo, and Adenilton J. da Silva, Quantum variational distance-based centroid classifier, Neurocomputing 576, 127356 (2024).
- Jungyun Lee and Daniel K. Park, Quadratic speed-ups in quantum kernelized binary classification, Adv. Quantum Technol. 7, 2400126 (2024).
- Patrick Rebentrost, Masoud Mohseni, and Seth Lloyd, Quantum support vector machine for big data classification, Phys. Rev. Lett. 113, 130503 (2014).
- Morten Kjaergaard, Mollie E. Schwartz, Jochen Braumüller, Philip Krantz, Joel I.-J. Wang, Simon Gustavsson, and William D. Oliver, Superconducting qubits: Current state of play, Annu. Rev. Condens. Matter Phys. 11, 369 (2020).
- Petar Jurcevic, Ali Javadi-Abhari, Lev S. Bishop, Isaac Lauer, Daniela F. Bogorin, Markus Brink, Lauren Capelluto, Oktay Günlük, Toshinari Itoko, Naoki Kanazawa, et al., Demonstration of quantum volume 64 on a superconducting quantum computing system, Quantum Sci. Technol. 6, 025020 (2021).
- Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C. Bardin, Rami Barends, Rupak Biswas, Sergio Boixo, Fernando GSL Brandao, David A. Buell, et al., Quantum supremacy using a programmable superconducting processor, Nature 574, 505 (2019).
- Jay M. Gambetta, Jerry M. Chow, and Matthias Steffen, Building logical qubits in a superconducting quantum computing system, npj Quantum Inf. 3, 2 (2017).
- Kenneth Wright, Kristin M. Beck, Sea Debnath, J. M. Amini, Y. Nam, N. Grzesiak, J.-S. Chen, N. C. Pisenti, M. Chmielewski, C. Collins, et al., Benchmarking an 11-qubit quantum computer, Nat. Commun. 10, 5464 (2019).
- Juan M. Pino, Jennifer M. Dreiling, Caroline Figgatt, John P. Gaebler, Steven A. Moses, M. S. Allman, C. H. Baldwin, Michael Foss-Feig, David Hayes, Karl Mayer, et al., Demonstration of the trapped-ion quantum CCD computer architecture, Nature 592, 209 (2021).
- Pascal Scholl, Michael Schuler, Hannah J. Williams, Alexander A. Eberharter, Daniel Barredo, Kai-Niklas Schymik, Vincent Lienhard, Louis-Paul Henry, Thomas C. Lang, Thierry Lahaye, et al., Quantum simulation of 2D antiferromagnets with hundreds of Rydberg atoms, Nature 595, 233 (2021).
- Loïc Henriet, Lucas Beguin, Adrien Signoles, Thierry Lahaye, Antoine Browaeys, Georges-Olivier Reymond, and Christophe Jurczak, Quantum computing with neutral atoms, Quantum 4, 327 (2020).
- Simon J. Evered, Dolev Bluvstein, Marcin Kalinowski, Sepehr Ebadi, Tom Manovitz, Hengyun Zhou, Sophie H. Li, Alexandra A. Geim, Tout T. Wang, Nishad Maskara, et al., High-fidelity parallel entangling gates on a neutral-atom quantum computer, Nature 622, 268 (2023).
- Michael A. Nielsen and Isaac L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge University Press, New York, 2011), 10th ed.
- Heinz-Peter Breuer and Francesco Petruccione, The Theory of Open Quantum Systems (OUP, Oxford, 2002).
- Ali Javadi-Abhari, Matthew Treinish, Kevin Krsulich, Christopher J. Wood, Jake Lishman, Julien Gacon, Simon Martiel, Paul D. Nation, Lev S. Bishop, Andrew W. Cross, Blake R. Johnson, and Jay M. Gambetta, Quantum computing with Qiskit, arXiv preprint arXiv:2405.08810
- F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay, Scikit-learn: Machine learning in Python, J. Mach. Learn. Res. 12, 2825 (2011).
- R. A. Fisher, Iris. UCI Machine Learning Repository, 1936. https://doi.org/10.24432/C56C76.
- Stefan Aeberhard and M. Forina, Wine. UCI Machine Learning Repository, 1992. https://doi.org/10.24432/C5PC7J.
- E. Alpaydin and C. Kaynak, Optical Recognition of Handwritten Digits. UCI Machine Learning Repository, 1998. https://doi.org/10.24432/C50P49.
- Terry Sejnowski and R. Gorman, Connectionist Bench (Sonar, Mines vs. Rocks). UCI Machine Learning Repository, 1988. https://doi.org/10.24432/C5T01Q.
- Jack W. Smith, Using the ADAP learning algorithm to forecast the onset of diabetes mellitus, in Proceedings of the Annual Symposium on Computer Application in Medical Care (IEEE Computer Society Press, 1988), pp. 261–265.
- N. A. Rodríguez-Briones and D. K. Park, qDNA-yonsei. GitHub Repository, 2025. https://github.com/qDNA-yonsei/BQR4QML.
